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4PM1/01R Jun 2020, Question 10: Solving Trig Equations

Pearson Edexcel International GCSE Further Pure Mathematics 4PM1 · Paper 1R · Calculator · 11 marks

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The question, in short

Solve three trigonometric equations in given intervals: one in x with π terms, one in θ to nearest degree, one in y over a negative degree range.

Summarised in our own words. Use the official paper for the exact wording and any diagrams.

Part (a) · 3 marks

Solve sin(x + π/3) = √3/2 for 0 ≤ x ≤ 2π, giving answers in terms of π.

  1. 1π/3 ≤ x + π/3 ≤ 7π/3
  2. 2x + π/3 = π/3, 2π/3, 7π/3
  3. 3x = 0, π/3, 2π

Answerx = 0, π/3, 2π

Part (b) · 3 marks

Solve 3sin θ + 5cos θ = 0 for −360° ≤ θ ≤ 360°, giving answers to the nearest degree.

  1. 13sin θ = −5cos θ ⇒ tan θ = −5/3
  2. 2Principal value θ = −59.04…°
  3. 3Add multiples of 180°: θ = −239°, −59°, 121°, 301°

Answerθ = −239°, −59°, 121°, 301°

Part (c) · 5 marks

Solve 1 + sin 2y = 2cos²2y for −180° ≤ y ≤ 0°.

  1. 11 + sin 2y = 2(1 − sin²2y)
  2. 22sin²2y + sin 2y − 1 = 0
  3. 3(sin 2y + 1)(2sin 2y − 1) = 0 ⇒ sin 2y = −1 or 1/2
  4. 4−360° ≤ 2y ≤ 0°: 2y = −90°, −210°, −330°
  5. 5y = −45°, −105°, −165°

Answery = −45°, −105°, −165°

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